Nearly-complete Decomposability and Stochastic Stability with an Application to Cournot Oligopoly

Nearly-complete Decomposability and Stochastic

Stability with an Application to Cournot Oligopoly

Jacco Thijssen†

April 2005

Abstract

This paper presents a general framework for analysing stochastic stability in models with evolution at two levels. Under certain conditions the theory of nearly-complete decomposability can be used to disentangle these two levels. They can then be studied separately and the equilibrium of one can be used to obtain the equilibrium of the other. This gives an approximation of the equi-librium of the combined dynamics. This approached is applied to a model of conjectural variation and imitation in Cournot oligopoly. If behavioural change takes place infrequently, the Walrasian equilibrium is the unique stochastically stable outcome. As a corollary, it is indicated that smaller industries are more competitive than larger ones.

Keywords: Stochastic stability, Near-complete decomposability, Oligopoly, Con-jectural variations, Imitation

JEL classification: L13, C73

∗_{I thank Dolf Talman for his meticulous proof reading. Helpful comments and suggestions from}

Sem Borst, Peter Kort, Marieke Quant, and seminar participants at Tilburg University, Universita’ d’Annunzio (Pescara), and the EARIE annual meeting 2001 in Dublin are greatly appreciated. Part of this research was carried out when the author was at the CentER for Economic Research, Tilburg University, Tilburg, the Netherlands. The usual disclaimer applies.

1

Introduction

In recent years, many papers and books have used the concept of stochastic stabil-ity to explain equilibrium selection in games. Most of these models follow the same structure.1 One starts by modelling a particular kind of dynamic behaviour in dis-crete time that evolves according to a Markov chain. This gives the pure dynamics. One can think, for example, of symmetric Cournot oligopoly where at each point in time firms imitate the output of the firm with highest profits in the previous period. Vega–Redondo (1997) shows that this dynamics has numerous equilibria (absorbing states), namely all those states where all firms produce the same quantity (the so-calledmonomorphic states). In order to select between all these equilibria, the pure dynamics is then perturbed by random noise, leading to the perturbed dynamics. In the oligopoly example, one assumes that with a certain small probability each firm chooses a random quantity. The stochastically stable states are those states that get positive probability mass in the limit distribution2 as the random noise component vanishes. In the oligopoly example Vega–Redondo (1997) shows that the Walrasian equilibrium is the unique stochastically stable state, thus giving an evolutionary underpinning of Walrasian behaviour.

Stochastic stability has been analysed to study, for example, evolution in biology (e.g. Foster and Young (1990)), the evolution of conventions (e.g. Young (1993)), equilibrium selection in non-cooperative games (e.g. Kandori et al. (1993)), and a plethora of other fields. In oligopoly theory, stochastic stability has been applied in the seminal Vega–Redondo (1997) as outlined above. This model has been ex-tended to study, for example, entry and exit (Al´os-Ferrer et al. (1999)), Bertrand competition (Al´os-Ferrer et al. (2000)) the comparison between Cournot and Wal-rasian equilibrium (Al´os-Ferrer (2004)), and the interaction between different types of behaviour (e.g. Schipper (2003) and Kaarbøe and Tieman (1999)).

One crucial assumption in all these models is that agents may change their de-cisions, but never the behaviour that leads to these decisions. In Vega–Redondo (1997), for example, all firms are profit imitators. Even if there are multiple be-havioural rules present in the population (as in e.g. Schipper (2003)), players can-not change their behaviour. This is a very restrictive assumption. One would like to be able to study models where agents can choose between different behavioural rules.3 _{In a repeated non-cooperative game, this would lead to two levels of}

dynam-1_{For a textbook exposition see, for example, Fudenberg and Levine (1998).}

2_{The limit distribution is also called the “invariant probability measure”, or “equilibrium}

dis-tribution”. The latter term can be confusing as the limit distribution need not correspond to an equilibrium of the underlying game.

ics. Given a configuration of behavioural rules for all players there is a dynamics of strategy choices, the strategy dynamics. At a higher level of aggregation, given the results of behavioural rules, players switch between different behavioural rules, the

behavioural dynamics. The question then is how the two levels influence each other and what behaviour and strategy combinations result in stochastically stable states. A major problem with such an analysis is that the resulting Markov chain de-scribing the dynamics becomes very complicated. This paper discusses a very intu-itive way of disentangling the two types of dynamics and, hence, obtaining a good approximation of the limit distribution of the original Markov chain. This approach is basically an application of the theory of nearly-complete decomposability as de-veloped by Ando and Fisher (1963), Simon and Ando (1961), and Courtois (1977). Originally, this theory was developed to aggregate over large dynamic systems in the presence of limited computational power. The main idea is that under certain conditions one can study the two levels of dynamics separately. Intuitively, this means that one uses the limit distribution of the strategy dynamics to obtain the limit distribution of the behavioural dynamics. The theory of nearly-completely de-composable systems provides an upper bound on the interaction between the two dynamics below which the latter limit distribution is a good approximation of the limit distribution of the original Markov chain.

To illustrate the way this theory can be used, we study an extension to Vega– Redondo (1997). We consider a Cournot oligopoly with a finite number of identical firms. The strategy dynamics is driven by best responses given conjectural vari-ations. For each configuration of conjectural variations this dynamics leads to a different equilibrium. In particular, there are configurations that lead to the Wal-rasian, Cournot-Nash, and Cartel equilibria. At the behavioural level it is assumed that firms imitate the behaviour (i.e. the conjectural variation) of the firm with the highest profit. It is shown that if behavioural change does not take place too frequently (a notion made precise below), Walrasian behaviour is (approximately) the unique stochastically stable state. This result reaffirms the strength of the Walrasian idea in competitive markets. A crucial assumption, however, is that be-havioural change takes place at a sufficiently low rate. The upper bound on this rate is decreasing in the number of firms. If behavioural change takes place more frequently, the support of the limit distribution may consist of more elements than just the Walrasian equilibrium, indicating that larger industries may inherently be less competitive than smaller industries.

The remainder of the paper is organised as follows. In Section 2 the general

framework is discussed. The theory of nearly-completely decomposable systems is described Section 3. In Section 4 we develop a model of Cournot oligopoly to illustrate the theory and Section 5 concludes.

2

A General Model of Multi-level Evolution

Let³In,(Si)i∈In,(ui)i∈In

´

be a game in normal form. Let S=Q

i∈InSi. This game

is infinitely repeated. At every time t = 0,1,2, . . ., the configuration of strategy choices is denoted by s_t = (s_it)i∈In, where sit ∈Si is the strategy chosen by player

iat timet. It is assumed that every player i∈In can choose from a finite setBi of

behavioural rules. A behavioural rule is a correspondenceBi :S →Si, with typical

element sit = Bi,t−1(st−1), where Bi,t−1 denotes the behavioural rule that player

i ∈ In chose at time t−1.4 That is, Bi,t−1(st−1) describes the strategy choice of

player iat time t, given the strategy choices of all players and player i’s choice of behavioural rule at timet−1.5 _{Furthermore, the fact that strategy choice at timet}

is influenced by behavioural choice at time t−1 suggests that strategy choices are made at the beginning of each period, whereas behavioural choices are made at the end.

It is assumed that players adapt their strategy choice with probabilityp∈(0,1) every period. So, with probability p, sit∈Bi,t−1(st−1), and with probability 1−p,

sit = si,t−1. If Bi,t−1(st−1) contains more than one element, player i chooses an

element at random according to a probability measure, ηi, with full support. The

aforementioned dynamics describe thepure strategy dynamics.

The actual strategy choice can be influenced by several aspects. For example, a player can make a mistake and choose another strategy than her behavioural rule prescribes. Another possibility is that a player experiments and consciously chooses another strategy. Finally, a player may be replaced by another player who has the same behaviour, but chooses a different strategy at first. Since these effects are outside the model, they are treated as stochastic perturbations. To model these perturbations at the strategy level, it is assumed that with probability ε > 0 a player chooses an element from S randomly according to a probability distribution,

νi, with full support.

4_{This formulation should not necessarily be interpreted as a player choosing a behavioural rule}

out of free will. The possibility that choice of behaviour is determined by, for example, genetics or evolutionary forces is nota priori excluded.

5_{This formulation does not explicitly include memory longer than one period. Models can,}

however, relatively easily be transformed to include finite memory. See, for example, Al´os-Ferrer et al. (1999).

Let s(k), k = 1, . . . , m, and BI, I = 1, . . . , N, denote the k-th and I-th per-mutation of S and B = Q

i∈InBi, respectively. Then, pure strategy dynamics and

perturbations, together, lead to a Markov chain onSwith transition matrixM_Iε(k, l), a typical element of which is

M_Iε(k, l) = Y i∈In ½ (1−ε)hp11¡ s(l)i∈BI i ¡ s(k)−i ¢η_i(s(l)_i) + (1−p)11¡ s(k)i=s(l)i ¢ i +ενi ¡ s(l)i ¢ ¾ ,

where 11_(·)denotes the indicator function and the part between square brackets gives the transition probabilities for the pure strategy dynamics.

The behavioural dynamics takes place at the end of periodt, when each playeri

gets the opportunity to revise its behaviour with probability 0<p <˜ 1. Behavioural change can be thought of as a conscious or non-conscious change. In the case of conscious change one could think that once in a while a player analyses her past performance and assesses the payoffs her behaviour yield by comparing with the payoffs of the other players. The importance of relative payoffs has already been stressed by Alchian (1950). It is assumed that each player knows the model and can observe the choices of other players and can, therefore, deduce the behaviour of the other players as well. She can then change her behaviour accordingly. Since deriving the other players’ behaviour requires more cognitive effort than simply following a behavioural rule in choosing a strategy, it seems reasonable to assume that players change their behaviour less often than their strategy choices which could be reflected in assuming that ˜p < p. Non-conscious behavioural change can, for example, be thought of as genetic change, where “weaker genes” are replaced by “stronger” genes, leading to a Darwinian survival of the fittest. Again, it seems reasonable to assume that behavioural change takes place at a lower frequency than strategy change.

For each player i ∈ In, behavioural change is governed by a correspondence

˜

Bi : B ×S → Bi. That is, given the strategy choices (s1,t−1, . . . , sn,t−1) ∈ S and

the behavioural rules (B1,t−1, . . . , Bn,t−1) ∈ B, player i chooses a behavioural rule

at timet such that

Bit ∈B˜i¡s1,t−1, . . . , sn,t−1;B1,t−1, . . . , Bn,t−1¢.

If ˜Bi(·) does not consist of a unique element, playerichooses any element from ˜B(·)

using a probability measure ˜ηi(·) with full support.

the strategy dynamics, random perturbations are added.6 _{So, each player chooses}

with probability ˜ε >0 any behavioural rule using a probability measure ˜νi(·) with

full support. For each k∈ {1, . . . , m} and corresponding strategy vector s(k), the behavioural dynamics gives rise to a Markov chain on B with transition matrixλε_k˜, a typical element of which equals

λ˜ε_k(I, J) = Y i∈In ½ (1−ε˜)hp˜11¡ BJ i∈B˜i(B I_,s(k))¢η˜i(B J i) + (1−p˜)11¡ BJ i=B I i) ¢ i + ˜εν˜i¡BiJ) ¢ ¾ , (1)

where the part between square brackets gives the transition probabilities for the

pure behavioural dynamics.

The combined strategy and behavioural dynamics yield a Markov chain onS× B

with transition matrixQε,ε˜. For future convenience, it is assumed that entries in this transition matrix are grouped according to the behavioural index. So, the k-th row inQε,˜ε consists of the transition probabilities from the state with behavioural rules

B1 and strategies s(k). Similarly, the m(I −1) +k-th row contains the transition probabilities from the state with behavioural rulesBI and strategiess(k). A typical element ofQε,˜ε is given by

Qε,ε˜(kI, lJ) =MIε(k, l)λεk˜(I, J),

which should be read as the transition probability form the state with behavioural rules BI and strategies s(k) to the state with behavioural rules BJ and strategies

s(l). Note that, for every ε >0 and every ˜ε >0, this Markov chain is ergodic (i.e.

Qε,˜ε is irreducible) and, hence, has a unique limit distribution. Consider the Markov chain with transition matrix

Q= lim

˜

ε↓0limε↓0Q

ε,˜ε_.

This is the transition matrix of the Markov chain that results after the stochastic perturbations at the strategy and behavioural level have converged to zero, respec-tively. The order of taking limits is essential to the analysis. The stochastically stable states are defined to be those states that have positive probability mass in the limit distribution, µ(·), ofQ. The dynamics in Qis complicated due to the in-teraction between the strategy and the behavioural levels. Therefore, the techniques developed in Freidlin and Wentzell (1984), which are usually used to determine the stochastically stable state will, in general, be hard to apply.

6_{In the case of non-conscious behavioural change, these perturbations can be thought of as}

One way of proceeding is to decompose the Markov chain Q. Let for every

I = 1, . . . , N, Q∗_I = limε↓0MIε. The standard techniques as applied in most of the

literature (cf. Young (1998)) can be used to the unique limit distribution of Q∗_I, denoted by µI(·). Then, construct a Markov chain on B, ˜Q, where µI(·) is used to aggregate over the strategy dynamics. That is the transition probabilities in ˜Q

are based on the assumption that the strategy dynamics has settled in equilibrium. Standard techniques can then be used to obtain the limit distribution, ˜µ(·) of ˜Q. The theory of nearly-complete decomposability gives conditions on when ˜µ(·) is a good approximation of the measure of interest, µ(·).

3

Nearly-complete Decomposability

Intuitively, a nearly-completely decomposable system is a Markov chain where the matrix of transition probabilities can be divided into blocks such that the interaction

between blocks is small relative to interaction within blocks. This section presents the formal theory.7 In the remainder, letQbe ann×nirreducible stochastic matrix, representing, for example, the transition matrix of an ergodic Markov chain. The dynamic process (yt)t∈IN, whereyt∈IRn for allt∈IN, is then given by

(yt+1)>= (yt)>Q. (2)

Note that Q can be written as follows:

Q=Q∗+ζC, (3)

where Q∗ is of order nand given by

Q∗=        Q∗₁ 0 . . . 0 0 . .. ... ... .. . . .. ... 0 0 . . . 0 Q∗_N        . (4)

The matrices Q∗_I, I = 1, . . . , N, are irreducible stochastic matrices of order n(I). Hence, n=PN

I=1n(I). Therefore the sums of the rows ofC are zero. We choose ζ

and C such that for all rows kI,I = 1, . . . , N,k= 1, . . . , n, it holds that

ζX J6=I n(J) X l=1 CkIlJ = X J6=I n(J) X l=1 QkIlJ (5)

and ζ = max kI ³ X J6=I n(J) X l=1 QkIlJ ´ , (6)

where the kI denotes thek-th element in the I-th block. The parameter ζ is called

themaximum degree of coupling between subsystems Q∗_I.

It is assumed that all eigenvalues of Q and Q∗ are distinct. Then the spectral composition8 of the t-step probabilities – Qt – can be written as

Qt= N X I=1 n(I) X k=1 λt(kI)Z(kI), (7) where Z(kI) =s(kI)−1v(kI)v(kI)>,

λ(kI) is the kI-th maximal eigenvalue in absolute value of Q, v(kI) is the

corre-sponding eigenvector normalised to one using the vector normk · k1, ands(kI) is the

condition number s(kI) =v(kI)>v(kI). Since Qis a stochastic matrix, the

Perron-Frobenius theorem gives that the maximal eigenvalue of Qequals 1. Therefore, (7) can be rewritten as Qt=Z(11) + N X I=2 λt(1I)Z(1I) + N X I=1 n(I) X k=2 λt(kI)Z(kI). (8)

If one defines for each matrixQ∗_I in a similar wayZ∗(kI),s∗(kI),λ∗(kI), andv∗(kI),

e.g. λ∗(kI) is thek-th maximal eigenvalue in absolute value ofQ∗_I, then one can find

a similar spectral decomposition for Q∗, i.e. (Q∗)t= N X I=1 Z∗(1I) + N X I=1 n(I) X k=2 (λ∗)t(k_I)Z∗(k_I), (9) using the fact that v_k∗

I(1I) = n(I)

−1 _{for all k}

I. The behaviour through time of yt

and y∗_t, where the dynamics of (yt)t∈IN is described by (2) and the process (yt∗)t∈IN

is defined by

(y_t∗₊₁)>= (y∗_t)>Q∗,

are, therefore, also specified by (8) and (9). The behaviour ofytcan be seen as

long-run behaviour whereas y_t∗ describes short-run behaviour. The comparison between both processes follows from two theorems as stated by Simon and Ando (1961).

Theorem 1 For an arbitrary positive real number ξ, there exists a numberζξ such that for ζ < ζ_ξ, max p,q |Zpq(kI)−Z ∗ pq(kI)|< ξ,

for any 2≤k≤n(I), 1≤I ≤N, where 1≤p, q≤n.

Theorem 2 For an arbitrary positive real numberω, there exists a numberζω such

that for ζ < ζ_ω,

max

k,l |ZkIlJ(kI)−v

∗

lJ(1J)αIJ(1K)|< ω,

for any 1 ≤k≤n(I), 1≤l ≤n(J), 1 ≤K, I, J ≤N, and where αIJ(1K) is given

by αIJ(1K) = n(I) X k=1 n(J) X l=1 v∗_kIzkIlJ(1K).

It can be shown that for allI = 1, . . . , N,λ(1I) is close to unity. Thereforeλt(1I)

will also be close to unity for small t. Hence, the first two terms on the right-hand side of (8) will not vary much fort < T2, for someT2 >0. The first term of the

right-hand-side of (9) does not change at all. Hence, for t < T₂ the behaviour through time ofytandyt∗is determined by the last terms ofQtand (Q∗)t, respectively. Also,

if ζ → 0 it can be shown thatλ(kI) → λ∗(kI) and from Theorem 1 it follows that

Z(kI) → Z∗(kI), for all k = 2, . . . , n(I) and I = 1, . . . , N. This means that for ζ

small and t < T2 the paths ofyt and y∗t are very close.

The eigenvalues λ∗(k_I) are strictly less than unity in absolute value for allk= 2, . . . , n(I), and I = 1, . . . , N. For any positive real number ξ1 we can therefore

define a smallest timeT₁∗ such that max 1≤p,q≤n ¯ ¯ ¯ N X I=1 n(I) X k=2 (λ∗)t(k_I)Z_pq∗ (k_I) ¯ ¯ ¯< ξ1 fort > T ∗ 1.

Similarly, we can find aT₁ such that max 1≤p,q≤n ¯ ¯ ¯ N X I=1 n(I) X k=2 λt(k_I)Zpq(kI) ¯ ¯ ¯< ξ1 fort > T1.

Theorem 1 plus convergence of the eigenvalues with ζ then ensures thatT1 →T1∗ as

ζ →0. We can always choose ζ such that T2 > T1. As long as ζ is not identical to

zero it holds thatλ(1I) is not identical to unity for I = 2, . . . , N.9 Therefore, there

will be a timeT3 >0 such that for sufficiently smallξ3,

max 1≤p,q≤n ¯ ¯ ¯ N X I=1 n(I) X k=2 λt(1I)Zpq(1I) ¯ ¯ ¯< ξ3 fort > T3.

This implies that forT2 < t < T3, the last term ofQtis negligible and the path ofyt

is determined by the first two components of Qt. According to Theorem 2 it holds that for any I andJ the elements of Z(1K),

ZkI1J(1K), . . . , ZkIlJ(1K), . . . , ZkIn(J)J(1K),

depend essentially on I, J and l, and are almost independent of k. So, for any I

and J they are proportional to the elements of the eigenvector ofQ∗_J corresponding to the largest eigenvalue. Since Q∗ is stochastic and irreducible, this eigenvector corresponds to the limit distributionµ∗_J of the Markov chain with transition matrix

Q∗_J. Thus, forT2 < t < T3 the elements of the vectoryt, (ylJ)t, will approximately

have a constant ratio that is similar to that of the elements ofµ∗_J. Finally, fort > T₃

the behaviour of yt is almost completely determined by the first term of Qt. So, yt

evolves towards v(11), which corresponds to the limit distribution µ of the Markov

chain with transition matrix Q. Summarising, the dynamics of yt can be described

as follows.

1. Short-run dynamics: t < T1. The predominant terms in Qt and (Q∗)tare the

last ones. Hence, yt and y∗t evolve similarly.

2. Short-run equilibrium: T1 < t < T2. The last terms of Qt and (Q∗)t have

vanished while for all I, λt(1I) remains close to unity. A similar equilibrium

is therefore reached within each subsystem of Qand Q∗.

3. Long-run dynamics: T2 < t < T3. The predominant term in Qt is the second

one. The whole system moves to equilibrium, while the short-run equilibria in the subsystems are approximately maintained.

4. Long-run equilibrium: t > T₃. The first term of Qt dominates. Therefore, a global equilibrium is attained.

The above theory implies that one can estimate µ(·) by calculating µ∗_I for I = 1, . . . , N, and the invariant measure ˜µof the process

(˜yt+1)>= (˜yt)>P, (10)

where (˜yI)t =Pn_k=1(I)(ykI)t for all I = 1, . . . , N, and some transition matrix P. For

t > T2 we saw that (ykI) t (˜yI)t ≈µ

∗

I,k. Hence, the probability of a transition from group

I to group J is given by (p_IJ)t+1 = (˜yI)−t1 n(I) X k=1 (y_kI)t n(J) X l=1 Q_kIlJ.

For t > T2 this can be approximated by (pIJ)t+1 ≈ n(I) X k=1 µ∗_I,k n(J) X l=1 QkIlJ ≡pIJ. (11)

So, by taking P = [p_IJ], the process in (10) gives a good approximation for t > T₂

of the entire process (yt)t∈IN.

Until now we have not been concerned by how largeζ can be. It was stated that for T₁∗ < t < T2, the original system Q is in a short-run equilibrium close to the

equilibrium of the completely decomposable system Q∗. If this is to occur it must hold that T₁∗ < T₂. Every matrix Q can be written in the form of (3), but not for all matrices it holds that T₁∗ < T2. Systems that satisfy the condition T1∗ < T2 are

called nearly-completely decomposable systems (cf. Ando and Fisher (1963)). Since

T₁∗ is independent ofζ and T2 increases with ζ → 0, the condition is satisfied for ζ

sufficiently small.

The main results concerning nearly-complete decomposability are given in the theorem below.

Theorem 3 (Courtois (1977)) Let Q be an irreducible stochastic matrix, with a decomposition as given in (3)-(6). If ζ < 1₂h1− max I=1,...,N|λ ∗₍₂ I)| i , (12)

thenQis nearly-completely decomposable. Furthermore, the limit distribution of the Markov chain with transition matrix P, as defined in (11) gives an O(ζ) approxi-mation of the limit distribution of Q.

4

An Application: Cournot Oligopoly with Conjectural

Variations and Imitation

In this section, an application of the theory of nearly-completely decomposable sys-tems to stochastic stability with multi-level evolution is discussed. Let be given a dynamic market for a homogeneous good withnfirms, indexed byIn={1,2, . . . , n}.

At each point in time, t ∈ IN, competition takes place in a Cournot fashion, i.e. by means of quantity setting. Inverse demand is given by a smooth function

P : IR+ → IR+ satisfying P0(·) < 0. The production technology is assumed to be

the same for each firm and is reflected by a smooth cost function C : IR+ → IR+,

satisfying C0(·) >0 and either C00(·) >0 orC00(·) <0. If at time t∈IN the vector of quantities is given byq ∈IRn₊, the profit for firm i∈I_n at time tis given by

where q−i= (qj)j6=i and Q−i =Pj6=iqj.

Each firm i∈ In chooses quantities from a finite grid Γi. Define Γ = Q_i∈InΓi.

For further reference let q(k), k = 1, . . . , m, be the k-th permutation of Γ. It is assumed that in setting their quantities firms conjecture that their change in quantity results in an immediate change in the total quantity provided by their competitors. This can also be seen to reflect the firm’s conjecture of the competitiveness of the market. Formally, firm i∈ I_n conjectures a value for the partial derivative of Q_−i

with respect to qi. Using this conjecture, the firm wants to maximise next period’s

profit. Hence, the firm is a myopic optimiser, which reflects its bounded rationality. The first-order condition for profit maximisation of firmireads

P0(qi+Q−i) ³ 1 +∂Q−i ∂qi ´ qi+P(qi+Q−i)−C0(qi) = 0. (13)

As can be seen from (13) we assume that there is only a first order conjecture effect. Furthermore, we assume that this effect is linear. These assumptions add to the firm’s bounded rationality.10

To facilitate further analysis, the conjectures are parameterised by a vectorα∈

IRn such that for alli∈I_n

(1 +αi)

n

2 = 1 +

∂Q−i

∂q_i .

Given a vector of conjectures an equilibrium for the market is given byq∈IRn₊such that for all i∈ I_n the first-order condition (13) is satisfied. Note that if all firms

i ∈ In have the conjecture αi =−1, the equilibrium coincides with the Walrasian

equilibrium. Furthermore, if all firms have αi = 2−_nn or αi = 1, the equilibrium

coincides with the Cournot-Nash equilibrium or the cartel equilibrium, respectively. Therefore, the conjectures αi = −1, αi = 2−_nn, and αi = 1 will be called the

Walrasian, Cournot-Nash, and cartel conjectures, respectively.

Each firm chooses its conjecture from a finite grid Λ on [−1,1], where it is assumed that Λ⊃ {−1,2−_nn,1}. The bounds of this finite grid represent the extreme cases of full competition (α = −1) and cartel (α = 1). For further reference, let

α(I),I = 1, . . . , N, be theI-th permutation of Λn=Q i∈InΛ.

This model is a special case of the general framework presented in Section 2. The pure strategy dynamics is such that firm i ∈ In seeks to find qit ∈ Γi so as

to approximate as closely as possible the first-order condition (13). That is, q_it ∈

10_{The first-order and linearity assumptions are also made throughout the static literature on}

conjectural variations. This seems incompatible with the assumption of fully rational firms in these models.

Bi(q₋t−_i1;αt_i−1), where11 forq−i ∈Qj6=iΓj andαi∈Λi, B(q−i, αi) = arg min q∈Γi n¯ ¯ ¯P 0_(q+Q −i)(1 +αi) n 2q+P(q+Q−i)−C 0_(q)¯¯ ¯ o .

The pure behavioural dynamics consists of firms imitating at timetthe conjec-ture of the firm(s) with the highest profit in periodt−1.12 Formally, firmi’s choice

αt_i is such thatαt_i ∈B˜(αt−1, qt), where for given α∈Λn _{and q∈Γ,}

˜ B(α, q) = arg max γ∈Λ n ∃j∈In:αj =γ,∀k∈In :π(qj, q−j)≥π(qk, q−k) o .

The combined strategy and behavioural dynamics yield a Markov chain on Γ×Λn with transition matrixQε,ε˜. A typical element of Qε,ε˜is given by

Qε,ε˜(kI, lJ) =MIε(k, l)λεk˜(I, J),

which should be read as the transition probability form the state with conjectures

α(I) and quantitiesq(k) to the state with conjectures α(J) and quantities q(l). We want to determine the unique limit distribution, µ(·), of the Markov chain with transition matrix

Q= lim

˜

ε↓0limε↓0Q

ε,˜ε_.

First the strategy dynamics is studied. As before, for eachI = 1, . . . , N, letQ∗_I = lim

ε↓0M

ε

I be the limit Markov chain when the perturbations in the strategy dynamics

vanish. Note that MI has a unique limit distribution, µI(·). To facilitate further

analysis it is assumed that for any vector of conjectures there is a unique equilibrium, i.e. a unique vector of quantities that solves (13) for all firms. Furthermore, we assume that this equilibrium is an element of the quantity grid Γ.

Assumption 1 For all α∈Λn_{there exists a unique q}α_{∈Γsuch that for alli∈I} n, P0(q_iα+Qα_−i)(1 +αi) n 2q α i +P(qiα+Qα−i)−C0(qαi) = 0.

Let the permutation on Γ that corresponds to qα be denoted byk(I), i.e. q(k(I)) =

qα. The following proposition states that for each vector of conjectures α(I) the unique stochastically stable state of the strategy dynamics is given byqα(I).

Proposition 1 LetI ∈ {1, . . . , N} be given. Under Assumption 1, the unique limit distribution µI(·) of the Markov chain with transition matrixQ∗_I is such that

µI(qα(I)) = 1.

11_{From the definition of behavioural rules, α}

i is not an argument of Bi. We include it for

clarification.

Proof. The proposition is proved using the theory developed by Milgrom and Roberts (1991). First note that for all i∈In, Γi is a compact subset of IR+. Define

for all i∈In the (continuous) function ˜πi : IR+×IRn+−1→IR+, given by

˜ πi(qi, q−i) =− ¯ ¯ ¯P 0_(q i+Q−i)(1 +αi(I)) n 2qi+P(qi+Q−i)−C 0_(q i) ¯ ¯ ¯.

Consider the normal-form game³In,(Γi)i∈In,(˜πi)i∈In

´

. LetS⊂Γ, denote bySi the

projection ofSon Γiand defineS−i=Q_j6=iSj. For alli∈Inthe set of undominated

strategies with respect toS is given by the set

Ui(S) = n qi ∈Γi ¯ ¯ ¯∀y∈Si∃q−i∈S−i : ˜πi(qi, q−i)≥˜πi(y, q−i) o . LetU(S) =Q

i∈InUi(S), thek-th iterate of which is given byU

k_{(S) =U}³_Uk−1_(S)´_,

k≥2, where U1(S) =U(S). Note that since qα(I) is unique we have

U∞(Γ) ={qα(I)}.

Following Milgrom and Roberts (1991) we say that{qt}t∈IN isconsistent with

adap-tive learning if ∀_tˆ_∈IN∃¯_t>tˆ∀˜t≥¯t:q ˜ t_∈U¡ {qs|ˆt≤s <t˜}¢ .

Let ˆt∈IN, take ¯t= ˆt+ 1 and let ˜t= ¯t+kfor somek∈ {0,1,2, . . .}. Then

{qs|ˆt≤s <˜t}={qs|s= ˆt, . . . ,¯t+k−1}.

Let{qt}t∈IN be generated by the pure strategy dynamics, i.e. the strategy dynamics

without the random perturbations. Then we have, by definition, that

∀y∈Γi : ˜πi(q ˜ t i, q ˜ t−1 −i )≥π˜i(y, q−˜t−i1).

Furthermore, it holds that q˜t−1 ∈ {qs|¯t ≤ s < ˜t}. Hence, we can conclude that

{qt}t∈IN is consistent with adaptive learning. From Milgrom and Roberts (1991,

Theorem 7) one obtains that kqt−qα(I)k →0 as t→ ∞. Since Γ is finite we have

∃¯t∈IN∀t≥¯t:qt=qα(I).

So, {qα(I)} is the only recurrent state of the pure strategy dynamics. From Young (1993) we know that the stochastically stable states are among the recurrent states of the mutation-free dynamics. Hence, µI(qα(I)) = 1. ¤

Before we turn to Proposition 2, the following lemma is introduced, which plays a pivotal role in its proof. It compares the equilibrium profits for different conjectures. Suppose that the market is in a monomorphic state, i.e. all firms have the same conjecture. The question is what happens to equilibrium profits if k firms deviate

to another conjecture. Ifn−k firms have a conjecture equal to α andk firms have a conjecture equal to α0, let the (unique) equilibrium quantities be denoted by q_kα

and q_kα0, respectively.

Lemma 1 For allk∈ {1,2, . . . , n−1} andα > α0 ≥ −1 it holds that

P¡

(n−k)qα_k +kq_kα0¢

qα_k0−C(qα_k0)> P¡

(n−k)qα_k +kqα_k0¢

qα_k −C(qα_k).

The proof of this lemma can be found in Appendix A. Lemma 1 plays a similar role as the claim in Vega–Redondo (1997, p. 381). The main result in that paper is driven by the fact that if at least one firm plays the Walrasian quantity against the other firms playing another quantity, the firm with the Walrasian quantity has a strictly higher profit. In our model the dynamics is more elaborate. Suppose that all firms have the Walrasian conjecture and that the strategy dynamics is in equilibrium, i.e. the Walrasian equilibrium. If at least one player has another conjecture not only its own equilibrium quantity changes, but also the equilibrium quantities of the firms that still have the Walrasian conjecture. Lemma 1 states that the firms with the lower conjecture still have the highest equilibrium profit. This is intuitively clear form the first-order condition (13). The firms with the lower conjecture increase their production until the difference between the price and the marginal costs reaches a lower, but positive, level than the firms with the higher conjecture. Therefore, the total profit of having a lower conjecture is higher. This happens because the firms do not realise that in the future their behaviour will be imitated by other firms which puts downward pressure on industry profits.

Some additional notation and assumptions are needed in the following. For a matrix A let λj(A) denote the j-th largest eigenvalue in absolute value of A.

Furthermore, define λ_k(I, J) = lim

˜

ε↓0λ ˜

ε

k(I, J) and let ζ = max_k

I n P K6=I Pm l=1QkIlK o . The following assumptions are made.

Assumption 2 All eigenvalues of Q are distinct.

Assumption 3 ζ < 1₂h1− max I∈{1,...,N}λ2(Q ∗ I) i .

Since the probability measures νi(·) and ˜νi(·) have full support for all i ∈ In, all

eigenvalues ofQwill generically be distinct and, hence, Assumption 2 will generically be satisfied. Letα(1) be the monomorphic state where all firms have the Walrasian conjecture, i.e. α(1) = (−1, . . . ,−1) We can now state the following proposition.

Proposition 2 Suppose that Assumptions 1–3 hold. Then there exists an ergodic Markov chain on Λn with transition matrix Q˜ and unique limit distribution µ˜(·). Forµ˜(·) it holds thatµ˜(qα(1)) = 1. Furthermore,µ˜(·) is an approximation ofµ(·) of

Proof. First, decomposeQas in (3)-(6). So, the transition matrixQis decomposed into a block diagonal matrixQ∗, where each diagonal block is the transition matrix for the strategy dynamics for a given vector of conjectures, and a matrix that re-flects the behavioural dynamics. The constantζ is the maximum degree of coupling between subsystems Q∗_I.

Given the result of Proposition 1 one can aggregateQusingµI(·) in the following way. Define a Markov chain on Λn with transition matrix ˜Q, which has typical element ˜ Q(I, J) = m X k=1 µI¡ q(k)¢ m X l=1 QkIlJ = m X k=1 µI¡ q(k)¢ λk(I, J) m X l=1 MI(k, l) = m X k=1 µI¡q(k)¢ λk(I, J) =λk(I)(I, J).

Note that the transition matrix ˜Qis the limit of a sequence of ergodic Markov chains with transition matrices ˜Qε˜ with ˜Q˜ε(I, J) = λε_k˜_(I)(I, J). So, ˜Q has a unique limit distribution ˜µ(·). Under Assumptions 2 and 3, Theorem 3 directly yields that ˜µ(·) is anO(ζ) approximation of µ(·).

The result on ˜µ(·) is obtained by using the familiar techniques developed by Freidlin and Wentzell (1984). First we establish the set of recurrent states for the mutation-free dynamics of ˜Qε˜. This is the dynamics without the experimentation part and is thus equal for all ˜ε > 0. From (1) one can see that the transition probabilities for this dynamics are equal to the transition probabilities of going from one vector of conjectures α(I) to another vector α(J) given that the current quantity vector is the equilibriumqα(I). So, the dynamics of ˜Qε˜is the pure conjecture dynamics if the quantity dynamics gets sufficient time to settle in equilibrium. Let the transition matrix for this aggregated pure conjecture dynamics be denoted by

˜

Q0.

Lemma 2 The setA of recurrent states for the aggregated mutation-free conjecture dynamics with transition matrix Q˜0 is given by the set of monomorphic states, i.e.

A=©

{(α, . . . , α)}¯ ¯α∈Λ

ª

.

The proof of this lemma can be found in Appendix B. Define the costs between

α(I) andα(J) to be c(α(I), α(J)) = min K=1,...,N © d(α(I), α(K))¯ ¯Q˜₀(K, J)>0 ª ,

where d(α(I), α(K)) = P

i∈In11¡_{αi(I)6=αi(K)}¢. The cost between α(I) and α(J) is

the minimum number of mutations fromα(I) that is needed for the pure conjecture dynamics to have positive probability of reaching α(J). Let α∈Λn_{. An α-treeH}

α

is a collection of ordered pairs (α0, α00) such that:

1. everyα0 ∈Λn\{α}is the first element of exactly one pair;

2. for all α0 ∈Λn\{α} there exists a path (α0, α1),(α1, α2),. . .,(αs−1, αs), (αs, α) inHα.

For eachα-treeHα the cost of treeHα is defined by

c(Hα) =

X

(α0_,α00_)∈Hα

c(α0, α00).

First, we build anα(1)-treeH∗with minimal costs. Then it is shown that for any stateα ∈ A\{α(1)} and any α-treeH_α the costs will be higher. From Freidlin and Wentzell (1984, Lemma 6.3.1) one can then conclude thatα(1) is the unique element in the support of ˜µ(·). Young (1993) has shown that the minimum cost tree is among theα-trees whereαis an element of a recurrent class of the mutation-free dynamics. Thus, from Lemma 2 we know that we only need to consider the monomorphic states in A. This implies that for all α-trees H_α,α ∈ A, we havec(H_α) ≥ |A| −1, since one always needs at least one experiment to leave a monomorphic state.

Consider α(1) and the α(1)-tree H∗ that is constructed in the following way. Let α ∈ A\{α(1)}. For all i ∈ In we have αi > αi(1). Suppose that one firm i

experiments to αi(1) = −1, while the other firms cannot revise their output.

Ac-cording to Lemma 1 withk= 1 this firm has a higher profit in quantity equilibrium than the other firms. If one period later all other firmsj 6=iget the opportunity to revise their conjectural variation (which happens with positive probability) they will all choose αj(1) =−1. Hence, one mutation suffices to reach α(1) and, therefore,

c(H∗) =|A| −1.

Conversely, let H_α be an α-tree for some α ∈ A\{α(1)}. Then somewhere in this tree there is a path between α(1) and a monomorphic state α0 with α0_i > −1 for all i∈In. Suppose that starting from α(1) one firm iexperiments to α0_i. From

Lemma 1 with k =n−1 it is obtained that firm ihas a strictly lower profit than the other firms in quantity equilibrium. So, to drive the system away from α(1) to

α0 at least two mutations are needed. Hence, c(Hα)> c(H∗). ¤

Proposition 2 gives a result on the convergence of market interaction to the Walrasian equilibrium that is similar to the result of Vega–Redondo (1997). Appar-ently, profit imitation is such a strong force that it also drives this more elaborate

Proposition 2 is an approximation. It might well be that the support ofµ(·) consists of more states than just the Walrasian equilibrium.

A crucial assumption is the one on the maximum degree of coupling between subsystems Q∗_I, ζ, as stated in Assumption 3. This parameter should not be too large. Intuitively, this condition requires that the interaction between subsystemsQ∗_I

is sufficiently low, i.e. that the conjecture dynamics does not happen too frequent. In Proposition 3 a sufficient condition on ˜p is given for Assumption 3 to hold.

Proposition 3 If p <˜ 1−¡₃

4

¢1/n

, then Assumption 3 is satisfied.

Proof. LetI ∈ {1,2, . . . , N}. From Bauer et al. (1969) we obtain an upper bound for the second largest eigenvalue in absolute value of Q∗_I:

λ₂(Q∗_I)≤minn max 1≤θ,ρ≤m 1 2 m X i=1 v_i1(Q∗_I)¯¯ ¯ Q∗_I(i, θ) v1_θ(Q∗_I) − Q∗_I(i, ρ) v1 ρ(Q∗I) ¯ ¯ ¯, max 1≤θ,ρ≤m 1 2 m X i=1 |Q∗_I(θ, i)−Q∗_I(ρ, i)|o, (14)

wherev1(Q∗_I) is the eigenvector corresponding to the largest eigenvalue ofQ∗_I. Since

Q∗_I is a stochastic matrix we have that

v_i1(Q∗_I) =µI_i = 11_(q=qα(I)₎.

Consider the first term on the right-hand side of (14). For θ=k(I) andρ 6=k(I), we get 1 2 m X i=1 v_i1(Q∗_I)¯¯ ¯ Q∗_I(i, θ) v1_θ(Q∗_I) − Q∗_I(i, ρ) v1 ρ(Q∗I) ¯ ¯ ¯= 1 2 ¯ ¯ ¯ Q∗_I(k(I), θ) µI_k(I) − Q∗_I(k(I), ρ) µI ρ ¯ ¯ ¯. (15)

Take θ = k(I), and ρ 6= k(I) such that q(ρ)i = q(k(I))i for some i ∈ In. Then

Q∗(k(I), ρ)>0 andµI_ρ= 0, so that (15) is unbounded.

The maximum of the second term on the right-hand side of (14) is attained for

θ=k(I) and some ρ6=k(I), such thatq(k(I)) is not a best response to q(ρ). One obtains that 1 2 m X i=1 |Q∗_I(k(I), i)−Q∗_I(ρ, i)| ≤ 1 2|Q ∗ I(k(I), k(I))|= 1 2,

since q(k(I)) is a best response toq(k(I)). Hence, we find thatλ2(Q∗_I) ≤ 1₂ for all

I = 1, . . . , N. So, we have that 1 2[1−I=1max,...,Nλ2(Q ∗ I)]≥ 1 4.

Note that it holds that ζ = max kI n X K6=I m X l=1 QkIlK o = max kI n X K6=I λk(I, K) o = max kI n 1−lim ˜ ε↓0λ ˜ ε k(I, I) o .

Furthermore, by definition we have that

λε_k˜(I, I)≥ Y i∈In

{(1−ε˜)(1−p˜) + ˜εν˜i(αi(I))}.

Therefore, we conclude that

ζ ≤1−lim ˜ ε↓0 Y i∈In {(1−ε˜)(1−p˜) + ˜εν˜i(αi(I))} = 1−(1−p˜)n< 1 4 ⇐⇒ p <˜ 1− µ 3 4 ¶1/n ,

which proves the proposition. ¤

Note that this upper bound is exponentially decreasing in the number of firms, from ˜p < 0.14 for n = 2, to ˜p < 0.03 for n = 10. This implies that, for the Walrasian equilibrium to be the only stochastically stable state in a large industry, the rate of behavioural change needs to be very low. In other words, leaving the rate of behavioural change constant, this analysis indicates that smaller industries are possibly more competitive than larger industries, since uniqueness of the Walrasian equilibrium as the only stochastically stable state cannot be guaranteed.

5

Discussion

This paper analysed a general framework for analysing stochastic stability in models with multi-level evolution, namely strategy and behavioural evolution. It was shown that under certain conditions, the theory of near-complete decomposability can be used to disentangle the two levels of evolution. They can then be studied separately and the equilibrium of the strategy dynamics can be used to obtain the equilibrium of the behavioural dynamics, which, in turn, is an approximation of the equilibrium of the combined dynamics. This approach is applied to an extension of the Vega– Redondo (1997) model of imitation in Cournot oligopoly.

In the application, we model strategy dynamics based on myopic optimisation by firms that includes the conjectured market response to the firm’s own quantity-setting behaviour which is modelled by means of a conjecture parameter. At a second level, we allow firms to change or adapt their behaviour in the sense that they can change their conjecture. This decision is also modelled to be boundedly rational. Firms look at their competitors and imitate the behaviour of the most successful firm.

The main conclusion of Proposition 2 is that if behavioural adjustment takes place at a sufficiently low frequency, the market ends up in the Walrasian equilib-rium in the long-run. An explicit upper bound for this frequency is provided and is shown to be exponentially decreasing in the number of firms. So, even with more elaborate behavioural dynamics than e.g. Vega–Redondo (1997), evolution still se-lects the Walrasian equilibrium. The appeal of this equilibrium lies in the fact that if behaviour is guided by profit imitation, i.e. relative payoffs, this leads to spitefulness in a firm’s actions. This in turn leads to selection of the Walrasian equilibrium. The analysis indicates that smaller industries are possibly more competitive than larger industries.

Modelling explicit dynamic processes where players learn from the past is im-portant, since in a ”pure repeated game framework[...]history matters only because firms threaten it to matter” (Vives (1999)). With learning or evolution, history matters per se. Until recently, most models of learning are restricted to dynamics at one level. The analysis in this paper suggests ways in which to include several levels of evolution. This makes it possible to study both learning and evolution sep-arately in a unified framework, i.e. the short-run and the long-run. In the example of the oligopolistic industry: how do (short-run) strategic choices based on conjec-tures and (long-run) competitive and Darwinian pressures interact? We restricted ourselves to two levels, but in principle it is straightforward to extend the theory of near-complete decomposability to more levels of learning or evolution.

Another application of the presented theory is for simulation analysis. Analysing large dynamic agent based systems can be computationally very intensive. If one is willing to assume relatively infrequent interactions between different levels of dynamics, the theory of near-complete decomposability can greatly reduce the com-putational burden of these models.

Appendix

A

Proof of Lemma 1

Since all firms are identical and solutions to the first-order conditions are unique, firms with the same conjecture have the same equilibrium quantity. Therefore, the equilibrium quantities q_kα and q_kα0 satisfy13

P0¡ (n−k)qα_k +kqα_k0¢ (1 +α)n 2q α k +P ¡ (n−k)qα_k +kq_kα0¢ −C0(q_kα) = 0 P0¡ (n−k)qα_k +kq_kα0¢ (1 +α0)n 2q α0 k +P ¡ (n−k)qα_k +kq_kα0¢ −C0(q_kα0) = 0.

These first-order conditions imply that

P0¡ (n−k)qα_k +kqα_k0¢ (1 +α)n 2q α k −C0(qkα) =P0¡(n−k)q_kα+kqα_k0¢(1 +α0)n 2q α0 k −C0(qα 0 k ). (A.1)

Suppose that q_kα≥qα_k0. There are two possible cases:

1. if C0(qα_k)≥C0(q_kα0), then (A.1) immediately gives a contradiction; 2. if C0(qα_k)< C0(q_kα0), then according to (A.1) it should hold that

−P0¡(n−k)q_kα+kqα_k0¢(1 +α)n 2q α k ≤P0 ¡ (n−k)qα_k +kq_kα0¢(1 +α0)n 2q α0 k .

This implies that q

α k qα0 k ≤ 1+₁₊α_α0. However, since q α k qα0 k

≥1 and 1+₁₊α_α0 <1 this gives a contradiction.

According to the mean-value theorem there exists a q ∈(q_kα, q_kα0) such that

C0(q) = C(q

α0

k )−C(qαk)

qα_k0−qα_k ,

since the cost function is continuous. Furthermore, it holds that

C0(q)<max{C0(q_kα), C0(q_kα0)} ≤P¡ (n−k)q_kα+kqα_k0¢ ⇐⇒P¡ (n−k)q_kα+kq_kα0¢ q_kα0 −C(q_kα0)> P¡ (n−k)q_kα+kq_kα0¢ q_kα−C(q_kα),

which proves the lemma. ¤

13_{Here the assumption that α}0

≥ −1 is crucial. For if α > −1 and α0 < −1 the system of first-order conditions has no solution.

B

Proof of Lemma 2

Given a monomorphic state, the pure conjecture dynamics remains in the same monomorphic state with probability one. SoA ⊃ ©

{(α, . . . , α)}¯ ¯α∈Λ

ª

. Conversely, let α ∈ Λn\A. With positive probability all firms may adjust their conjecture and with positive probability all choose the same conjecture, leading to a monomorphic state. Hence, A ⊂© {(α, . . . , α)}¯ ¯α∈Λ ª ,

which proves the lemma. ¤

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